[MAP] Liouville's theorem and electromagnetic fields

[Yuri Alexahin]

Hi Kirk,

Thank you for digging out these interesting papers.
Of course the Poincare invariants remain the same no matter what momenta are used.
But this is not what we calculate from tracking or measurement data using standard definition.
So a clarification is still needed of what and how we should calculate.

Yuri

----- Original Message -----
From: Kirk T McDonald <kirkmcd at Princeton.EDU>
Date: Thursday, March 10, 2011 4:09 pm
Subject: [MAP] Liouville's theorem and electromagnetic fields
To: MAP List <map-l at lists.bnl.gov>
Cc: Kirk McDonald <kirkmcd at Princeton.EDU>


> Folks,
> 
> There is a technical question as to how we should be calculating 
> emittance for beams in electromagnetic fields.
> 
> The formal theory of Liouville’s theorem is clear that the invariant 
> volume in phase space is to be calculated with the canonical momentum
> gamma m v + e A / c
> and not the mechanical momentum m v.
> 
> This is awkward in two ways:
> 1.   We don’t always know the vector potential of our fields
> 2.   The vector potential is subject to gauge transformations, so 
> canonical momentum is not gauge invariant.
> 
> The second issue is disconcerting in that it suggests that phase-space 
> volume, and emittance, are not actually invariant  -- with respect to 
> gauge transformations.
> 
> Hence, it is useful to note a very old paper,
> W.F.G. Swann, Phys. Rev. 44, 233 (1933)
> which shows that the phase-space volume for a set of noninteracting 
> particles is the same whether or not the term e A / c is included in 
> the “momentum”.
> 
> This result has the consequence that phase-space volume (and 
> emittance) is actually gauge invariant – although the location of a 
> volume element in space space is gauge dependent.
> 
> ---------------
> This suggests that we could simply calculate emittances based only on 
> the mechanical momentum, and avoid having to worry about the accuracy 
> of our model for the vector potential.
> 
> Of course, our calculations are actually of rms emittance, which is a 
> better representation of the “ideal” emittance if the phase-space 
> volume is more “spherical”, and not elongated/twisted.
> 
> It could be that the shape of the phase-space volume is better for rms 
> emittance calculation if the vector potential, in some favored gauge, 
> is included in the calculation.....
> 
> --Kirk
> 
> PS  I have placed Swann’s paper as DocDB 560
> http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase
> user = ionization pass = mucollider1
> 
> See also the paper by Lemaitre that used Liouville’s theorem for 
> cosmic rays in the Earth’s atmosphere (using mechanical momentum).   
> This may well be the earliest paper about particle beams and 
> Liouville’s theorem.
> 
> PPS  Scott Berg notes that when one evaluates emittance at a fixed 
> plane in space, rather than at a fixed time, it is better to use the 
> “longitudinal” coordinates (E,t) rather than (P_z,z).
> 
> Is there any written reference that explains this “well known” fact?
> 
> How is this prescription affected by electromagnetic fields?
> 
> The vector potential of even a simple rf accelerating cavity has an 
> A_z component (which is zero on axis, but nonzero off it).
> http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf
> Note that the vector potential is nonzero outside the cavity, even 
> though the E and B fields are zero there!
> 
> Do we know how to include A_z in our longitudinal emittance calculations?
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